Flawed macroeconomic models lead to erroneous conclusions

I get a lot of queries about the difference between fixed and flexible exchange rates in terms of the options that each present a sovereign, currency-issuing government. I considered this question several times in the past. Many of those questions are pitched in terms of the basic macroeconomic framework for an open economy that appears in most mainstream macroeconomics textbooks, particularly those written in the 1970s, 1980s and 1990s. I am referring here to the Mundell-Fleming model which has been the mainstream staple for many years. The modern textbooks still teach these models but the exposition has evolved although remains deeply flawed. It seems that this conceptual framework is still used to make public comments along the lines that the US government is facing insolvency and that the euro remains the best monetary organisation for Europe. Those conclusions are as flawed as the model that spawns them. Flawed macroeconomic models lead to erroneous conclusions.
If you studied macroeconomics in the 1970s and 1980s, the so-called Mundell–Fleming model was a dominant framework. It persists in macroeconomics textbooks even today and was based on the work of two economists, Robert Mundell and J. Marcus Fleming in the early 1960s. The primary references are easy to locate via search engines.

The model incorporated the international sector into the standard IS-LM model that constituted the Neo-Keynesian consensus. The IS-LM model brought together the equilibrium conditions in the product market with money market equilibrium to derive the interest rate and equilibrium national income level.

I won’t explain it here – in the text book that Randy Wray and I are writing at present you will not find an IS-LM model included. It is a deeply flawed framework which involves students learning all the modern neo-liberal myths such as budget deficits drive up interest rates, central banks control the money supply (and implict money multiplier), investment flows are somehow dependent on current interest rates, and the rest of it.

The work of Mundell-Fleming model just adds an external sector to an already flawed model and so perpetuates most of the misunderstandings that arise when we consider the impact of capital mobility, exchange rate movements, current account balances and the rest of it.

The model sought to provide a framework to analyse the effectiveness of macroeconomic policy under different assumptions of capital mobility. It spawned what became known as open-economy macroeconomics and the mainstream theoretical developments were conducted within this framework for many years.

For a small open economy, the model claims that the local interest rate will be driven by the global interest rate and that national income, money supply and/or exchange rate has to adjust to ensure that parity is maintained. This is because one of the basic versions of the model assume that capital mobility is perfect (no frictions).

Under assumptions of perfect capital mobility, the model claimed that fiscal policy would only be effective if exchange rates were fixed. the model claimed that increasing government expenditure would drive up local interest rates (so the usual flawed crowding out arguments) which would attract capital inflow and lead to an appreciating local currency.

The strengthening exchange rate would undermine net exports (exports become more expensive than imports become cheaper) and this continues until the local interest rate is forced down to the world interest rate by the decline in real income. At that point, the net export loss fully offsets the fiscal stimulus.

The model shows that under fixed exchange rates, the central bank reacts to the pressure on the exchange rate caused by a fiscal expansion by selling local currency in exchange for foreign currencies in international markets which ensures the exchange parity is maintained.

The expansion of the domestic money supply offsets the rising interest rates associated with the fiscal expansion (so-called “accommodation” of the fiscal expansion) and as a result the fiscal stimulus is effective.

So students who study macroeconomics within this framework are biased towards believing that the external sector wipes out the fiscal capacity of a government under flexibile exchange rates unless there is capital rigidity and that fixed exchange rates provide room for fiscal policy to influence real GDP.

The claim is that the government’s budget deficit leads to an increase in the trade deficits (the so-called “Twin Deficits” hypothesis) emerged initially from the use of Mundell-Fleming IS-LM macroeconomic models.

The narrative should now be obvious – expansionary fiscal policy allegedly pushes up domestic interest rates and leads to a higher exchange rate which crowds out exports (the Twin Deficits Hypothesis is in the same camp as the crowding out hypothesis but adds exchange rate crowding out to interest rate crowding out).

The appreciated exchange rate also is conducive to higher imports because for constant foreign prices they are now cheaper in the local currency.

Some economists have recently considered this hypothesis again for the US (Kim and Roubini, 2008) give that a lot of the work was previously done on data that applied to the fixed exchange rate period.

They found that expansionary fiscal policy actually improves the current account and depreciates the real exchange rate (indicator of competitiveness) – which they call the twin divergence hypothesis.

They show that the budget deficits allow for higher private savings (because of their stimulatory impact on national income). They also claim that investment falls because the real interest rate is higher but I don’t find that part of their analysis compelling at all.

Other models suggest that rising domestic absorption (as aggregate demand rises and income rise) links the deficits to the current account balance rather than any interest rate or real exchange rate effect.

The overall conclusion that you draw from the literature, however, is confusion. No robust findings are a systematically available and many complicating factors are evident.The methods used in the empirical studies are questionable and the results highly ambiguous.

Central banks set interest rates which as Japan has demonstrated for two decades are not related in any way to movements in fiscal policy. The current period is also demonstrating that.

Australia demonstrated in the late 1990s and beyond that record budget surpluses were associated with record depreciations in the exchange rate. More recently our exchange rate has reached levels not seen for 35 years while the budget has gone back into substantial deficit.

I may write some more about this model although to advance the discussion further I have to become somewhat technical which is not a very advisable strategy for a popular blog to take.

The modern significance of this work was that it was used to justify the creation of the Eurozone. Mundell’s theory purports to define the conditions under which several countries should form a monetary union. It was a case of textbook theory, inapplicable to the real world, being applied to suit ideological ambitions. Yet the European policy makers used the OCA literature relentlessly to give “authority” to their decisions.

Mundell defined three conditions, which if in existence, would justify the formation of a monetary union as an OCA:

The countries should face common consequences if hit by a negative shock (no asymmetric shocks). So, for example, unemployment rates should be similar across the countries in the union;

There should be a high degree of labor mobility and/or wage flexibility within the group of countries.

There is a common fiscal policy that can transfer resources from better performing to poorly performing countries.

A comparison of this list of “theoretical conditions” with what emerged out of the Maastricht theory, which stipulated five conditions: convergence in interest rates, budget deficits, public debt, inflation rates and stable exchange rates prior to the formation of the union, leads to the conclusion that the EMU is not remotely like an OCA.

The Maastricht conditions reflected the neo-liberal ideology that was gathering pace at the time and was dominating policy makers. The EMU countries did not (and do not) satisfy the Mundell criteria for an OCA yet they still went ahead with the common currency which meant that each sovereign nation gave up their monetary policy independence and allowed the SGP to hamstring their fiscal sovereignty.

At the time, Mundell overlooked the discrepancies between his theory and the obvious reality and was a supporter of the Euro creation. He remains a supporter today despite it being obvious that the experiment is a grand failure which is inflicting devastating consequences on millions of Europeans.

In recent years, Mundell has claimed that the Eurozone would benefit from increased wage flexibility. So workers in recessed regions would reduce their wage demands and make it easier for firms to hire them. Allegedly, this would also lower prices of final goods and services produced in that region which improves their competitiveness.

So while the EMU is really a fixed exchange rate system imposed on a number of countries and so nominal exchange rate depreciation cannot occur, the mainstream orthodoxy continue to believe that real terms of trade changes can happen if relative unit labour costs change.

This is how Mundell claimed it would work. The recessed nations could improve their competitive by cutting wages and unit labour costs (a real rather than a nominal depreciation) which would attract firms into that region away from other regions in the union.

Of-course, this assumes that employment is driven by wage levels. If a nation such as Spain tried to cut real wages (engineering this would be difficult in itself) then there would be a huge drop in demand by Spanish workers. It is highly unlikely that the so-called “real depreciation” impacts would offset the local income effects on demand.

We are seeing that this approach is now being applied in various countries with the effects that I have been predicting for some years. Cutting wages and working entitlements might reduce the supply curve (lower unit costs) although that is not even guaranteed if productivity also collapses due to failure to invest in best practice technology and loss of worker morale, increased sabotage etc.

But this sort of austerity also impacts on the demand side and it is clear that the latter impact is dominating as the Eurozone contracts (with even Germany recording negative real GDP growth in the December 2011 quarter).

Robert Mundell was at it again yesterday except now he is commenting on fiscal solvency issues in the US. In this Bloomberg Interview – Mundell claimed that a “lack of fiscal discipline” in the US is “pushing the world’s largest economy toward solvency issues”.

I suppose he would say that given he has been given a Nobel Prize in economics for previously flawed work (noted above).

Bloomberg quoted Mundell as saying:

The public is looking for free lunches, and the political competition for votes makes the politicians offer them free lunches … That’s what gets us in to the difficulties of insolvency … You could have fiscal stimulus back in the day of Keynes, when the government was a small proportion of gross domestic product and there was no insolvency problem … You can’t just issue more bonds to pay for deficits and expect it to solve the employment problem … The United States is not in as bad a situation as Europe … but it’s getting that way.

It is hard to actually know where to start with this set of statements.

When we think in terms of “free lunches” the only way we can make any sense of that is in real terms. The US government, like all sovereign governments has no financial constraint and the marginal cost of issuing new dollars is as close to zero as you can get.

The opportunity cost of a government policy is not the numbers that appear in the financial statements, but rather the extra real resources that use as a result of the policy intervention. Typically, when private spending growth is strong, there will be trade-offs, because if the public sector desires use of a particular real resource then the non-government sector will be denied use.

However at times of excess capacity (machines and labour), the opportunity cost is is likely to be low, especially if the unemployed are receiving income support to maintain minimum consumption standards.

I would agree with Robert Mundell about the compromised state of the US polity at present, although my conclusions about the impacts of the ineffective Congress would be different.

In the current situation, the political impasse that has emerged over the last year or so is in fact a blessing to that nation. For if the growing Republican groundswell had been able to more fully capture the Congress then the US would have followed Europe and Great Britain down the fiscal austerity path.

This is not to say that the fiscal stance currently in place in the US is in any way appropriate. There is substantial need for more fiscal stimulus to ensure the current growth accelerates and eats into th entrenched unemployment queue.

However, the implication for Mundell is that the political impasse in the Congress is delaying fiscal austerity, which is pushing the US towards insolvency.

Unless the Congress voluntarily decides to ignore its nominal liabilities denominated in the currency that the US government issues under monopoly conditions, then there is never an issue of solvency with respect to the US (at the federal level).

The US government can always meet its financial obligations when denominated in US dollars. A currency issuer can never become financially insolvent in its own currency.

Modern Monetary Theory (MMT) also allows us to understand that a statement such as “issue more bonds to pay the deficits” is not a accurate description of the way in which a sovereign currency works, notwithstanding voluntary accounting arrangements at shuffle revenue from bond sales into government expenditure accounts.

These accounting arrangements obscure the essential flows that occur in a modern monetary economy.

A currency-issuing government borrows what it has already spent. The flow of funds used by bondholders to buy the bonds came from government spending in the past.

Governments do not issue debt in order to borrow, even though it appears that they do so and they even say they do so.

The fundamental principles that arise in a fiat monetary system are as follows.

The central bank sets the short-term interest rate based on its policy aspirations.

Government spending is independent of borrowing which the latter best thought of as coming after spending.

Government spending provides the net financial assets (bank reserves) which ultimately represent the funds used by the non-government agents to purchase the debt.

The “penalty for not borrowing” is that the interest rate will fall to the bottom of the “corridor” prevailing in the country which may be zero if the central bank does not offer a return on reserves.

Government debt-issuance is a “monetary policy” operation rather than being intrinsic to fiscal policy, although in a modern monetary paradigm the distinctions between monetary and fiscal policy as traditionally defined are moot.

Accordingly, debt is issued as an interest-maintenance strategy by the central bank. It has no correspondence with any need to fund government spending.

So what would happen if the government simply declined to issue any debt? First, spending would still occur in just the same way (mostly crediting bank accounts) and the flow of demand into the economy would run up against supply and support real output production (as long as the real constraint was not compromised).

Second, the central bank would have to use reserve support rates to ensure it could maintain a non-zero interest rate target.

Third, the non-government sector would have to create an alternative (but not risk-free) asset upon which to price risk. At present, they use government bonds to serve this purpose.

Fourth, the non-government sector would lose an income stream (in the form of interest payments on the government debt).

However, I agree that a government cannot just run “deficits and expect it to solve the employment problem”. Alternative deficit compositions will impact on employment levels in different ways and so the design of the fiscal intervention is also something that has to be thought out carefully.

But it remains true that a well-designed fiscal intervention (particularly if accompanied by employment guarantee) can ensure that anyone who wants to work is able to find a job, irrespective of the state of non-government spending growth.

The overall quality of employment might be inferior (for example, in terms of matching skills with vacancies) if a nation has a large employment guarantee pool – that is, if it achieves what I call “loose” full employment through the use of a Job Guarantee.

But then again, the jobs that are lost in the private sector during a recession are often dominated by low-skilled and low-paid jobs, which means that a public employment guarantee is unlikely to offer an inferior, overall employment mix.

That point, however, is getting us off the track.

It was also not the case that fiscal policy was unconstrained in the “day of Keynes”. Fiscal policy had to play a supportive role to monetary policy which was tied to maintaining the fixed exchange rate. External deficit nations were always subject to the “stop-go” growth outlook because the domestic economy (and the aggregate policy instruments) were hostage to need to defend agreed external parity.

So when fiscal policy pushed the economy too fast and imports rose relative to exports, the central bank was forced to run contractionary policy to reduce growth. Countries with persistent external deficits were always running up against domestic contraction as a result.

Under flexible exchange rates, monetary policy is freed from maintaining the parity and interest rates can be set by the central bank to encourage local investment. Fiscal policy is then free to target appropriate levels of domestic spending, particularly with the aim of advancing public purpose (full employment and equity).

Finally, there is no valid comparison that can be made between the US government and the member-states in the Euro, with respect to their relative government deficits. The US government uses its own currency, whereas the member-states within the euro are forced (by their own volition) to use a foreign currency (the Euro).

The latter governments face financial constraints on the spending (because they have to get the “foreign” currency from either taxation or bond-issuance).

So a statement such as the “United States is not in as bad a situation as Europe … but it’s getting that way” carries no meaningful content.

In the Bloomberg interview, Robert Mundell also re-affirmed his support for the Euro. He said:

It’s political glue inside Europe to keep it together — the euro is the best thing going for it since the creation of the common market … The end game is going to be deeper integration in Europe and more centralization of the fiscal authority.

So no matter the real costs to the humans involved, if all the disparate nations in Europe are finally brought under the same fiscal yoke then the mission has been accomplished.

How long will that take?

How long does youth unemployment have to be around 50 per cent (and rising) before the glue sets?

Robert Mundell gave an interview to the UK Telegraph (August 25, 2011) – Professor Mundell, euro, and ‘pessimal currency areas’ – where he elaborated on his continued support for the Euro even though the political leaders mis-used his concept of an OCA to ram through the flawed, and now, failed monetary system design.

The article says of Mundell that:

Some say he has bent the theory of “Optimum Currency Areas” (OCA) to justify combining the vastly different economies of Europe in monetary union – whatever this implies for freedom and democratic legitimacy.

If you read his own pioneering work on OCAs – “A Theory of Optimum Currency Areas” in the American Economic Review of 1961- it is hard to see how the eurozone can possibly qualify.

But it reports that in the Interview, Robert Mundell has no doubts that the creation of the EMU was correct. The article quotes him as saying:

We’re in the midst of a very big crisis because nothing has been done yet to convince the markets that there has been a fundamental change. To save Europe there has to be a move in the direction of shared government.

I agree with that conclusion as far as it goes. There are two options facing the EMU: (a) form a proper federation with a strong federal fiscal authority capable of offsetting negative asymmetric demand shocks with appropriate fiscal transfers; and (b) dissolve the experiment altogether and restore national currency sovereignty and floating exchange rates.

Mundell says his preferred option (Option (a)) “will not be easy” to achieve. However, he is in the German camp and considers that such an option would work if there was fiscal austerity maintained (balanced budgets). My interpretation of the viability of the first option is that it would require substantial “federal” budget deficits and quite diverse transfers of funds between member states at times of slack private spending.

Attempting to introduce Option (a) with a fiscal compact will just result in stagnation and devastation in some regions (current member-states) of the federation.

Mundell claims that:

The euro hasn’t done anything bad. The problem is lack of fiscal discipline. Countries like Greece, Portugal, and Ireland have been on a spending spree on entitlements …

As the writer of the article correctly points out:

… if you strip out Greece, the euro crisis is not caused by lack of fiscal discipline in any meaningful sense. That is a Wagnerian myth, much promoted by Chancellor Angela Merkel. Spain and Ireland never violated the Maastricht limits on deficits. Both ran a big budget surplus during the boom when monetary policy was kept loose to help nurse Germany through a mini-slump. Italy is running a primary budget surplus. Ireland came close to eliminating its public debt altogether. The real problem is that EMU stoked a private sector credit bubble. That proved the killer.

All correct and – the collapse of the private credit bubble (which was encouraged by neo-liberal deregulation and lack of oversight of the banking and broader financial system) was met by governments who use a foreign currency, were operating within strict fiscal rules (the Stability and Growth Pact) which were impossible to meet, given the scale of the private spending collapse and, which, forced them to adopt fiscal austerity to “obey” the political masters in Brussels, Frankfurt and Washington (IMF).

The problem is inherent in the design of the monetary system. The problem is the Euro.

Further, trying to rely on domestic deflation is to commit the southern states to an impossible chase. As long as Germany suppress real wages growth (which they have been doing for 10 or so years), nations like Greece and Portugal will have to be continually eroding real standards of living and still never become “competitive” in international trade within the Eurozone.

The solution is to restore the exchange rate flexibility of these nations and allow the real income losses inherent in their reduced competitiveness to manifest in higher priced imports. The advantage of this approach is that their exports become more attractive and the governments are able to concentrate on domestic initiatives that create local employment.

It is not all Shangri La! but it will be a much better long-term strategy than more or less permanent suppression of local living standards which will be required under the current monetary system, and which will require on-going financial support from the ECB.

The Telegraph article quotes Joseph Stiglitz who noted that (with reference to Argentina’s default in 2001):

There is life after default, and life after leaving a fixed exchange rate system …

Conclusion

The Mundell-Fleming framework, which underpinned open-economy macroeconomics as practised by mainstream economists. The framework reinforced the mainstream myths about the role of government in the economy, its options, and the impacts of those options.

One of the driving intellectual forces in the development of that paradigm was Robert Mundell.

The framework has been discredited over many years and is not a reliable basis on which to understand policy options available to a sovereign government which issues its own currency.

Curiously, while Robert Mundell also developed the concept of an optimal currency area he continues to support the Eurozone, despite the fact that it never satisfied the conditions that he specified in his academic work for an OCA.

In other words, he bent the conditions to suit his ideological preference for this sort of arrangement. At the basis of this is his bias towards fiscal austerity.

Mundell’s framework was never fully implemented and disregarded when the government decided to abandon a fixed rate system for a floating rate. Mundell always argued that a reserve currency must be the anchor for all other currencies and that a floating rate system would lead to wide fluctuations creating instability in global markets.

Thought it worth noting, as the Wiki article does too, that MMT’s beloved ancestor, Abba Lerner, concisely considered such OCA matters, e.g. in his Economics of Employment, long before Mundell. Just did it right, though, so no prize for you!

“There is a common fiscal policy that can transfer resources from better performing to poorly performing countries.”

Actually, if one compares ECB rates to Taylor based projections, its target rate matched the required for Euro-core and fell short for the Euro periphery. It was way low during the credit boom years and it should be negative now (which translates in a need for big fiscal deficits).http://www.frbsf.org/publications/economics/letter/2011/el2011-18.html

“This is how Mundell claimed it would work. The recessed nations could improve their competitive by cutting wages and unit labour costs (a real rather than a nominal depreciation) which would attract firms into that region away from other regions in the union.”

And here’s the IMF’s (!!) view on internal devaluation:
“Despite deep nominal declines in wages and pensions, real effective exchange rate depreciations have been regularly only modest due to only limited pass-through to prices (Baltic states, Argentina, Greece). Furthermore, private sector corporations are more likely to cut employment than to fully adjust wages, even in fairly flexible labor markets (Latvia). It also takes a long time for resources to shift from the non-tradable to the tradable sector, and both persistent skill mismatches and lack of increased investments in the tradable sector preclude full factor reallocation (former East Germany, Latvia). External adjustment therefore works predominantly through import compression rather than an expansion of exports—and oftentimes imports contract long before any real depreciation of the exchange rate.”http://www.imf.org/external/pubs/ft/scr/2012/cr1257.pdf pages 48-49

“The public is looking for free lunches, and the political competition for votes makes the politicians offer them free lunches … That’s what gets us in to the difficulties of insolvency …”

Mundell is showing his complete contempt for democracy in this statement. Everything would be okay, he is claiming, if only the people weren’t so stupid; if only the people would let the smart folks be in charge. This we find is the justification of dictators throughout time; all the way back to Plato’s philosopher kings. Put me in charge; I know best.

Popper in his “Open Society” was more enlightened. He understood that the true strength of democracy lay not in the people’s brilliance, but rather in the system’s superior ability to make (vote in) corrections once it was discovered that some particular course of action was in error. Only democracy could defuse the egos of oligarchy that insist on staying the course even when they are wrong, all for the sake of saving the faces of those in charge but with little other skin in the game or fat in the fire.

Which of course is where we are now with the Euro project. Gradually, the democracy that is hailed as one of its founding principles is being stripped away in favor of the oligarchs who, in spite of the continuous failures of their selected policies, keep insisting that they know best; that the people must not be allowed to interfere with anything so disrupting as a democratic election with real choices on their ballots.

And it continues still, as Susan George of the Transnational Institute pointed out recently in the “Rise of Neoliberal and Undemocratic Europe” on the new fiscal compact. Rather than admit that the current program is not working, they are completely striping away the peoples’ voices and the peoples’ constitutions, instead doubling down, insisting that the enlightened ones were right all along if only the patient were properly bled and the leeches were properly fed.

@Benedict@Large
“Mundell is showing his complete contempt for democracy in this statement. Everything would be okay, he is claiming, if only the people weren’t so stupid; if only the people would let the smart folks be in charge. This we find is the justification of dictators throughout time; all the way back to Plato’s philosopher kings. Put me in charge; I know best.”

I don’t disagree but it is a two sided coin, to give a specific example, on TV, an oppostion politician suggested targetted spending would be a good way of stimulating the economy & a government minister immediately chanted the mantra “you can’t spend your way out of debt crisis”, nothing else & the entire studio audience start clapping in approval.
Poppers idea only works if the voters realise a particular course of action was in error, if they go through bad times but still believe the alternative course of action would have been even worse then the “error” is never corrected. While Education has to be the solution, its tough going when the population has been brainwashed into believing a series of commandments or little mantras as absolute facts which trump any reasoned alternative argument.

Bob,
“While Education has to be the solution, its tough going when the population has been brainwashed into believing a series of commandments or little mantras as absolute facts which trump any reasoned alternative argument.”

I doubt Bill would endorse this, but one solution is to use the same techiques they use. A good thing to do is, instead of correcting them using “reasoned argument”, simply point out where the evidence does not back their arguement – then leave them hanging!

“Under assumptions of perfect capital mobility, the model claimed that fiscal policy would only be effective if exchange rates were fixed. the model claimed that increasing government expenditure would drive up local interest rates (so the usual flawed crowding out arguments) which would attract capital inflow and lead to an appreciating local currency.”

” . . . we find that (i) the output effect of an increase in government consumption is larger in industrial than in developing countries, (ii) the fiscal multiplier is relatively large in economies operating under predetermined exchange rate but zero in economies operating under flexible exchange rates; (iii) fiscal multipliers in open economies are lower than in closed economies and (iv) fiscal multipliers in high-debt countries are also zero.”

While you posted this reference to a SVAR study of fiscal multipliers (which I was aware of) are you satisfied yourself that it is an authoritative study?

In particular, do you understand the identifying restrictions on the SVAR (highly restricted Cholesky decomposition); the “special” way they treated military expenditure; the time and national invariance imposed; the conflation of monetary systems in the study where some nations are sovereign in their own currencies and others use a foreign currency (a difference not considered in the matrix); the particular time lags imposed on the identifying restrictions; the particular (curious) way they dealt with interpolating annual data to quarterly frequency; the sensitivity of their results to the particular method of de-trending used; the problems that limited availability of government investment data; and several other major issues?

The paper is not an authoritative study and given the number of manipulations of data and the identifying restrictions (which bias the results in particular ways) it would have been possible to get vastly different results with different restrictions and data set construction.

Dear bill,
Thanks for your reply.
“While you posted this reference to a SVAR study of fiscal multipliers (which I was aware of) are you satisfied yourself that it is an authoritative study?”
No.
However, as this ‘study’ is likely to be quoted as evidence that fiscal policy is ineffective, I drew attention to it to try elicit some critical comment to be used as evidence against the ‘study’. Thanks for that.

A small point. Unless I have missed something they say, on page 1, “We have gone to great lengths to ensure that only data originally collected on a quarterly basis is included (as opposed to interpolated based on annual data).”

Thanks for all the hard work that you put in to ensure a very interesting blog.